5950번 - Bottleneck

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문제

Farmer John is gathering the cows. His farm contains a network of N (1 <= N <= 100,000) fields conveniently numbered 1..N and connected by N-1 unidirectional paths that eventually lead to field 1. The fields and paths form a tree.

Each field i > 1 has a single one-way, exiting path to field P_i, and currently contains C_i cows (1 <= C_i <= 1,000,000,000). In each time unit, no more than M_i (0 <= M_i <= 1,000,000,000) cows can travel from field i to field P_i (1 <= P_i <= N) (i.e., only M_i cows can traverse the path).

Farmer John wants all the cows to congregate in field 1 (which has no limit on the number of cows it may have). Rules are as follows:

Time is considered in discrete units.

Any given cow might traverse multiple paths in the same time unit. However, no more than M_i total cows can leave field i(i.e., traverse its exit path) in the same time unit.

Cows never move *away* from field #1.

In other words, every time step, each cow has the choice either to

stay in its current field

move through one or more fields toward field #1, as long as the bottleneck constraints for each path are not violated

Farmer John wants to know how many cows can arrive in field 1 by certain times. In particular, he has a list of K (1 <= K <= 10,000) times T_i (1 <= T_i <= 1,000,000,000), and he wants to know, for each T_i in the list, the maximum number of cows that can arrive at field 1 by T_i if scheduled to optimize this quantity.

Consider an example where the tree is a straight line, and the T_i list contains only T_1=5, and cows are distibuted as shown:

Locn: 1---2---3---4 <-- Pasture ID numbers
C_i: 0 1 12 12 <-- Current number of cows
M_i: 5 8 3 <-- Limits on path traversal; field
1 has no limit since it has no exit